Semisimple Frobenius structures at higher genus

By Alexander Givental. 2000

298 citations recorded by the source corpus.

In the context of equivariant Gromov-Witten theory of tori actions with isolated fixed points, we compute genus g > 1 Gromov-Witten potentials and their generalizations with gravitational descendents. Both formulas, with and without descendents, are stated in a form applicable to any semisimple Frobenius structure and therefore can be considered as definitions in the axiomatic context of Frobenius manifolds. In (nonequivariant) Gromov-Witten theory, they become conjectures expressing higher genus GW-invariants in terms of genus 0 GW-invariants of symplectic manifolds with generically semisimple quantum cup-product.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c6933a2abe74eff45dfcf906cd04d7447a88a909

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure (2012, methodology): “The recipe of reconstruction of the matrix R from [17] gives at the origin the matrix R(ζ) = ∑∞ k=0Rkζ , where (5-10) Rk = (2k − 1)!!(2k − 3)!! 24kk! · ( −1 (−1)k+12ki 2ki (−1)k+1 )”
Symplectic geometry of Frobenius structures (2003, methodology): “• According to [23, 24], the machinery of fixed point localization and summation over graphs used by M.”
The Extended Toda Hierarchy (2003, background): “The crucial point in proving the validity of this conjecture in full genera is the Givental’s result on the Virasoro constraints for CP 1 [18, 19].”
Relations on Mbar_{g,n} via 3-spin structures (2013, unknown): “For a concise summary see [7, Section 1].”
The structure of 2D semi-simple field theories (2007, methodology): “Confirming a prediction of Givental’s [12], we will see that semi-simple CohFT’s are determined by their genus-zero part, the restriction to families of genus zero curves, save for an ambiguity related to the Hodge bundle.”
The spaces of Laurent polynomials, Gromov-Witten theory of ℙ1-orbifolds and integrable hierarchies (2008, background): “Assume that X has semi-simple orbifold quantum cohomology and denote by DBQHðXÞ the descendent potential corresponding to the semi-simple Frobenius structure ([16]).”